hp–Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains.

Abstract

In this article, we develop the a posteriori error estimation of hp–version discontinuous Galerkin composite finite element methods for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. Although standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. Computable bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp–adaptive refinement procedure will be presented.